Threshold and Hitting Time for High-Order Connectedness in Random Hypergraphs

Abstract

We consider the following definition of connectivity in k-uniform hypergraphs: Two j-sets are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. We determine the threshold at which the random k-uniform hypergraph with edge probability p becomes j-connected with high probability. We also deduce a hitting time result for the random hypergraph process -the hypergraph becomes j-connected at exactly the moment when the last isolated j-set disappears. This generalises well-known results for graphs.

The size of the giant high‐order component in random hypergraphs

“…Since obtaining this result, in [10] we have demonstrated how this lemma also plays a key role in establishing the threshold for j-connectedness in H k (n, p). Indeed we derive a stronger result: the hitting time for j-connectedness in the random k-uniform hypergraph process coincides with the moment when the last isolated j-set disappears.…”

“…Apart from being crucial in the proof of Lemma 4, the smooth boundary lemma (Lemma 9) is certainly also very interesting in its own right and as such a major contribution of this paper. In fact, Lemma 9 has already proven to be a powerful tool in a broader context, for instance in the analysis of the property of a random hypergraph being j-connected [10].…”

Vanishing of cohomology groups of random simplicial complexes

“…It is known (see e.g. [12,38,39]) that p 0 is the threshold for vertex-connectedness of the random (k + 1)-uniform hypergraph, that is whp p T = (1 + o(1))p 0 (Lemma 4.1).…”

“…and thus in particular p T > p − 0 . Observe that Lemma 4.1 is equivalent to p 0 being a sharp threshold for vertexconnectedness of the random (k +1)-uniform hypergraph, which follows for instance from [12] or [39] as a special case of each (see also [38] for a stronger result). The proof relies on standard applications of the first and second moment methods and is an easy generalisation of the graph case (see e.g.…”

“…A first natural generalisation for dimension k ≥ 1 is the random (k + 1)-uniform hypergraph G p = G(k; n, p) in which each (k + 1)-tuple of vertices forms a hyperedge with probability p independently. There are several natural ways of defining connectedness of G p , which have been extensively studied, including vertex-connectedness [4,5,7,8,29,39,40] and high-order connectedness (also known as j-tuple-connectedness) [12,13,14,28]. Another topic which has received particular attention is generalisations of the ℓ-core of a random graph (i.e.…”